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One of the celebrated formulas of Ramanujan is about odd zeta values, whichhas been studied by many mathematicians over the years. A notable extension wasgiven by Grosswald in 1972. Following Ramanujan's idea, we rediscovered aRamanujan-type identity for $\zeta(2k+1)$ that was first established byMalurkar and later by Berndt using different techniques. In the current paper,we extend the aforementioned identity of Malurkar and Berndt to derive a newRamanujan-type identity for $L(2k+1, \chi_1)$, where $\chi_1$ is the principalcharacter modulo prime $p$. In the process, we encounter a new family ofRamanujan-type polynomials and we notice that a particular case of thesepolynomials has been studied by Lal\'{i}n and Rogers in 2013. Furthermore, weestablish a character analogue of Grosswald's identity and a few moreinteresting results inspired from the work of Gun, Murty and Rath.

A new Ramanujan-type identity for $L(2k+1,\chi_1)$